Question

How long will it take to recover an investment of $10,000 at 10% interest rate under the following scenarios?

a. $2,400 annually

b. An increasing gradient of $1,125 per year

c. $200 monthly

d. Why are the answers to parts a, b, and c equal or different?

Show cash flow diagrams and necessary equivalent models.

Answer 1

The present value of the investment is $10000 and the interest rate is 10%

a) In the case of an annual payment of $2400
=NPER(10%,-2400,10000)
= 5.655183

b) In the case of gradient payment of $1125, we will have to equate the PV of each cash flow to $10000

Year	Payment	PV @ 10%
1	1125	1022.73
2	2250	1859.5
3	3375	2535.69
4	4500	3073.56
5	2430	1508.84
	Sum	10000.3

The payment for 5th year should be $5625 but to equate to the PV of $10000, we get payment of $2430 by trial and error method.
2430 / 5625 = 0.432

So the time period should be ( 4 + 0.432 = 4.432 ) years.

c) In the case of a monthly payment of $200, we will have to change the interest rate as ( 10% / 12 )
=NPER(10%/12,-200,10000)
= 64.94871 Months
= 5.4123 Years

All the answers are different because of the different variables. In the first case, the payment is constant and made annually but in the 2nd case, payment was variable and as it was increasing as the time goes os the payment period is shorter.
In the 3rd case, the payment mode is monthly and so that compounding has affected the period.